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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Finitely generated abelian group ： ウィキペディア英語版 Finitely generated abelian group In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements ''x''1, ..., ''x''''s'' in ''G'' such that every ''x'' in ''G'' can be written in the form :''x'' = ''n''1''x''1 + ''n''2''x''2 + ... + ''n''''s''''x''''s'' with integers ''n''1, ..., ''n''''s''. In this case, we say that the set is a ''generating set'' of ''G'' or that ''x''1, ..., ''x''''s'' ''generate'' ''G''. Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below. ==Examples== * The integers $\backslash left(\backslash mathbb,+\backslash right)$ are a finitely generated abelian group. * The integers modulo $n$, $\backslash left(\backslash mathbb\_n,+\backslash right)$ are a finitely generated abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group $\backslash left(\backslash mathbb,+\backslash right)$ of rational numbers is not finitely generated:〔Silverman & Tate (1992), (p. 102 )〕 if $x\_1,\backslash ldots,x\_n$ are rational numbers, pick a natural number $k$ coprime to all the denominators; then $1/k$ cannot be generated by $x\_1,\backslash ldots,x\_n$. The group $\backslash left(\backslash mathbb^$ *,\cdot\right) of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition and real numbers under multiplication are also not finitely generated.〔〔La Harpe (2000), (p. 46 )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finitely generated abelian group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.